Sommerfeld Fine-Structure Formula for Two-Body Atoms

For relativistic atomic two-body systems such as the hydrogen atom, positronium, and muon-proton bound states, a two-body generalisation of the single-particle Sommerfeld fine-structure formula for the relativistic bound-state energies is found. The two-body Sommerfeld bound-state energy formula is obtained from a two-body wave equation which is physically correct to order $(Z\alpha)^4$. The two-body Sommerfeld formula makes two predictions in order $(Z\alpha)^6$ for every bound state and every mass ratio. With $N$ the Bohr quantum number: (a) The coefficient of the $(Z\alpha)^6/N^6$ energy term has a specified value which depends only on the masses of the bound particles, not on angular quantum numbers; (b) The coefficient of the $(Z\alpha)^6/N^4$ energy term is a specified multiple of the {\em square} of the coefficient of the $(Z\alpha)^4/N^3$ energy term. Both these predictions are verified in positronium by previous calculations to order $(Z\alpha)^6$ which used second-order perturbation theory. They are also correct in the Coulomb-Dirac limit. The effect of the two-body Sommerfeld formula on calculations of muon-proton bound-state energies is examined.